There are problems with increasing property taxes on elderly folks on fixed incomes, the solution isn't to say "you never own any tax on the appreciated value" the solution is to say "you can pay the tax on the appreciated value when you sell your home" which is an insane liberal tax and spend policy practiced in the hippie haven of the... state of Texas.
Additionally, the lions gain of the benefits of prop 13 don't accumulate to the working class elderly living in appreciated homes, it goes to giant corporations like Walt Disney (yes, Disney Land is protected by prop 13), golf courses, extremely wealthy property investors, and the heirs of people who have owned nice homes and then died who then bequeath both their house AND their insane tax rate on to their children.